Saturday, January 25

Divide by the sum of the base-7 digits

Consider a number divided by the sum of its base-7 digits. If it goes evenly, repeat the procedure with the quotient. Continue thus until either the quotient arrived at is not evenly divisible by the sum of its base-7 digits, or the number 1 is reached.

There are only a finite number of positive integers that reach 1, the largest being

40090274859335788188225568958946114753008293347953175307173216438617812041555279623245865158433989020626979196127925376719536463066943202165158879123115753785079089282793640091903556833926655783997154594289161232116101044714579121240616475797816175718660050822593870727643983209393172239451564000476429967804164112577663514902049782611642638387652639351006276378991504088440953822881824506926245966514963040000736614161156758934690996704884832607639278528354732245854666907653859254346591907729278186995119626281069810539346524668166171231605013177562186320341362087115862268089449413417647638277642047209420699562205067887315497916485984047479489310211370627319356060094500301725713900105120910701197104253117436639212500902642314176176967754312051840268795232231564024125267411543199976525204927541708786679816758575657980667513081688484501372900276995079844047822503749108162774331711019656062212497878785698569208460217904682951905411288417291826551415716078000233552616827475207210341514285121555751575441134399283505320677313761929657695840782124531572332588442099528937144165250184289803073385821791845909284971333115533197924549237472741867079666309831088832176663994090643038292403450620469240039382786840375357650171845483239864881868376636340925946996646785706327788942019760843817358046420110165297873316665218667309986852968308151914980109861255816132030094899126866728713978090367542979716058225895429295194207297444409428114690224793718858105439405028365635014177006033191195106091973264238591041056591447688345347096425755587171730534338260827764462028242056916005480589853515765864089539689949490780229330184685785893827574420324884514104456847507112315527845624705715303582403385098824135955821678344596336056449221276334860009653229579919137731666143204865856974858549099713061735336708747418929594760677046206036537198447075597441193263702819022900470439134462698668788248803875258982520998673154040971723246000322690663588338780282130172653742214062320339642980651332437592943224171993054278504854359263546513111508293137452801227303396833467880111969018400836956919418265829867587385464514229131616091155350352359987344887853698108073516675773966599621889517183012061233924452768645462660214005099381105533323925286404042847666102614749030207217427375546428039974955185046587777342073373801527021929645708531510107354271378582391987786332608126321798272426997044018165624866770048171105808016488238188289801308544677802188474644539742390690842248428997401476399010954819656592067781778703291889757954679498302069401401589325018144280087599532278107367903151149649200575433317349581179662094712894052336941995935158089574682049242371135766303405317333617059918464786830296092873607723446043773160610547278645024176648057510413700798874223247283413540627128291798831684104289783812109519627995369714012993163971461872686244749458099411550580146503461601748857853191027042128009474550163763421442030011736634544836079453887866911132961282642546949388578768339835845624161668674021819489723044668294972996997423875362731882339528027245606149737913469101024504136549552849759917083630366539409605504659561011785440345272778211465945693960720896567621969969082811639238016393851093246600903247987707991789982893268370340914432740301848602399384133541309133004765045945724615545504560252430675440785241940929800637869330405417472594113223551904980864884198100540690717524636636379084134321939117603083808987373406802510361535170295926711324895678515256387358372611315751426038086161408174212693132659421605854959177851387175215247272630481582104275216142647681484992006265256385073823101449946755442330251434072611011489208591071525519165345150834473188181963251496464787239046879191176057202681823756066807911350570696571416365892638067852245320918704458345943824312320158356113528232806739654094813862171549232903287378512320904267042814472764239592657055353795135816252806115279123777834378600455934649535749621601986287045846352331849132536698413203087525707996888614483678382817496804464755695026786614809413548315054727775622002491893507360222933168981350098069602745544880692844548161914242691954481967644658745708028218170823474907840566117715397438721084547595421235809009128516284771247412686929297166747924273183595558256149187553094692988422792885679295229828731648430259944575276876535059284962602025651564247674254961641303327067136820343570606305572710794009532899041484237145056396481252948208121293195293955066499814095083227206365565524478997083638007853145132760121614350459055718294903921968755626688776131622821943262038510531087337993411469959845109033850015746028838780928421937726067934065224646757252627968948602024554240345188812056557019930438014843762808997245846764508623318834940804266521012417189436145906314777397243820562742540006134033037797890593215167461055509964419803703129965420774258361409290750622155072090212298443017813234103132127967993050253187957756924572283399296498652057583661385290455882219424975377399318373244931565633364877586069108072644714859683638688770424939379193467063544688987844624228624247205955441197735403439459887097450104020193684565175047546198582440115835268718123063858197162132127571015278632833778510683979780534184475292335187466278947401076763810662399750205184968078564813253753559514481268830591427801643087836170980870565146296479969566879950986431989117581520467409581186824798648001708940283184463764630034743405896647862838818430708764981297899258485589028613929261169040191749894038083335510033454827457798132396634744308570096568006644461861208943095923030329281356018755810734378377890722570794284697707418194294698953112364072732053821462968057013810305937663419273519779308262603635181031709204706975137864123673673389746652359298764223256795650322005135012106617227408487832344213063757386025891329337543613990584854719922545294782142232296720520631751605489285364119476851574815545641853016892858351444380277340844050292111869425152072650972437614156199194581797893546437816320452146569512073424871341424421995845821366420601533448902732135660793193527473582840561865399234530347032833676527441445583718035934771654689666787986539415791379147997538937615589553377468156111114645881350167183786268670123661018578431393319437168263711324426050397765263367207253382559661860463697090083074542325857182683992721745520542403903747926889485124768140390471147892041301178940056769365494197657203060671809981976740860893858496811981329645117977446917074954385200549230224659873260454126053650211959542931832195809291013462962544694891228644468879102993507245061922720884257306136941735971027027897323720141951876732293091968290602574595066772225301644505395740366854399364361907936673806540051759442142234433808837456364413923460109598187266592396953273164025977573095883035930683609122387311759871669020849502221120675091612073626584261302003523531553137481158186131172626241101515098402454610021116809425957914212179101605181332671739827844671552277586007535053774283817581712197064353741549242785088859153239236838430990178303886234536118744437642497389754250755439371878096184678793025092321871455039500040305824307674862544485216813154870493294977549636437183966080364600262876101172520314461053033693239000742910781266762715316011336592993823705279006274004988895977499235007814639679237526355807064291823107834257782806922188435272733938760338024695343442669419544529844039516530458106118852146854958325173661000125143008498241984522423685248053771352436576577262153071086800388885154314639546535113040814564761013808760111497216000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
written in base ten, or

16305135035545012466241561026121103363306322322236102306112633635125110360151360300311453551142635324265123415554443332454022032233040544430302554231060412532131112116345613044565322211534161103334664051630552566011014222051234236045203546536102314012235205531252342040061531632412331533551053220253051042420303303631113423100254556460110245204303334064650116505101360330566353412030645146104452415500005624356214333051641130221406221044403240542552550505000615400036613432030451361156030555203112252452566623660115010134004236130161116132166222266226525103166320522165254423325655645022643554552444261054126542565444003545532543131430214300146534156632045511116226643015040424200211165641135432555523132040423340361361151660303666221621641554602134622206442464365446230411454064025141603105322622062410351152226562234402256363212225353600253032644551452536354666531145423132120154251652151531035660054000344162665655205203464635155112221451231233510460456454045245225442350165611065531445253504645431014210445143660632044225663663613002061512035436665010435233041360015423663245553425445264154343260346013522530416265146330236445646253312653456001242113424553465066662663360210036605406120006644633462201306442453434563333426630660123245140446642320125055454055205216225264013113020024053650405331523360515525452041456643161044323501532210305633461010214244520134636262461232065614560246605663006204514633201615401310252665561356346563152446235101525043256033626226401005651615343204154204135546015566024331521405205320064010613016336355641252104243110316664356364111662226114420012341400636011445610650464554625136262421230454436150265062454201533221003156643411256403401425020611310446630150003461316202253163462506023021252254122151121605441222260521261245111245413155615361535215644003010343454611600331530102134561441442543140603140450423612620301461200145214126143646033154553550520625211423441662420653621551436266335430160062102511153653365232105246354222311164030461512625556605366441135604333502641444524140126654025134425435061226064064605166224263502302614463232534451240113360413352302522113621011454612006020014364360526331254611540644532524042004502264055633530600503134236213460332252662222204415132665436264351331556516455445631031422323620650052404350535610035060320111166016240001203045021465232226616663221424155131314051601010551663063421233145214334560426041235302363123531402134466603152553330302052565443402435262354655223003164011533034635305240660454516622410633653453355422450623403415452166303145010623346223561023003304564403346462362352331231645620450360054216601156363615240245116545553503122150053052632163001601322156145316551653302334113562362534345352342321302623354362443000003606465341151244263322430210452041360422365116202153356162506211236365102162450642104253354011211331351303316636244654253226261260633641332414604545416036215664522611133412202160161003351040616143211531111122005102620333453233164461122233446205306415562644562165431165565036626150204363603662303460000225352050505016136466042545563461310306536335061524124304643502022555643015540665423612206324450430306442065120341653310226564043311146202052205246060465204264026063224054003421155536231024545435221603152454605214346466451530004006505456200026413612210601161504620303600061302210414234632633452426513036051553042256314243660441462505033065153035626062556113563264661136351046056443365200403415404234603413262540125360046306012360312102322206236346304025530654511560255653154420430426365255221233141325430456632553644636151363403605132131143640323635215556405646222122166502331532511556103663414063435110325554101304421240134646151411234116556513002332622303410066534312124514616060251063244646266131510314132606020663221362552533245646242160254135104564214356545510613430561035264665130663506153152240222403536251563255665323010125134232351444610440256232350433113332614662236035555104320663263550654262404266211540531025031105305626233622326015640225412230556453043212654335006614306610401205022330042312624465643441306146220153602523013323361513066030544364350220614116662064261616260334466214103233542036105203463603514551646361235012141363245132414141042004450520536160550150122605565115514306152446316115516331034300555063523331045305505163044363361114261462405506666515625134240005621016251015120111562141253663415325442065041313636421363643561126261142526166315664063340466232311025245435066033162456234405062030650112643154422526651254311332555402520226053206121252300614254604232124056556625220051142214045263213216316310430013246064460215136214445114563343050423644262624506124261403552015331104600545100011135455112152325162461346004160023363550050011465240650426245220565310536343126643215365523342220654315322125463610554222636522131346621221066102622565565302413201503304345411361035531365311555010300550360653243360504016226045360043311324164640412650041230462510120230366165505135634141233133400045230332415336662120214055355541345505640521053310330635016046403524621162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written in base seven.

This number takes 2046 steps to reach 1. I have plotted the count of positive integers (out of a total 693633 available) that take n steps (from n = 0 to n = 2046) to get to 1 in a graph. The smaller, darker-blue plot at the bottom-left is the count of positive integers (out of a total 15095 available) that take n steps (from n = 0 to n = 440) to get to 1 in the base-10 version of the procedure.

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